Mathematics Stack Exchange
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Questions tagged [euclidean-geometry]

For questions on geometry assuming Euclid's parallel postulate.

The geometry of Euclid is based on five axioms (Euclid called them postulates). Any geometry based on the first four of these is called an absolute geometry. The fifth one states:

If a line segment intersects two straight lines forming two interior angles on the same side that sum to less than two right angles, then the two lines, if extended indefinitely, meet on that side on which the angles sum to less than two right angles.

It was observed by Proclus that, in the presence of the the other four postulates, Euclid's fifth postulate can be replaced by Playfair's axiom:

Given a line and a point not on it, then one and only one line parallel to the given line can be drawn through the point.

The independence of the parallel postulate and its equivalent formulations from the first four axioms was shown by Beltrami in 1868.

Another alternative definition is that two lines are parallel if every perpendicular extended from one meets the other as a perpendicular.

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In the following figure prove that: $AK||BC$

In arbitrary triangle $ABC$, let $G$ be centroid and $AH$ be a height.We extend $HG$ beyond $G$ to intersect circumcircle at $K$.Prove that $AK||BC$ I connected midpoint of $BC$ to $K$ and $A$ to show $MA=MK$ but failed...
Hamid Reza Ebrahimi
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Is the sum of the diagonals of an isosceles trapezoid at least the sum of the bases?

Here's one question that has been bothering me now for a while. It is not homework. For an isosceles trapezoid (wikipedia link: http://en.wikipedia.org/wiki/Isosceles_trapezoid), do we always have that sum of the two diagonals is larger (or equal)…
T. Eskin
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Prove two circles can meet at most once on each side of their line of centers.

Honestly, I am not even sure what formation the question is referring to. I am guessing that if the circles meet only at one point, then that point is a point that lies on the straight line that is formed between the centers of two circles. Is my…
user3000482
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Please help me understand the definition of straight line given by Euclid.

"A straight-line is (any) one which lies evenly with points on itself." This is how Euclid defines a straight line but I don't know what it really means. Is this saying that any point picked on the straight line will have equal distance from each…
user3000482
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How many unique distances are there in a 5 x 5 grid?

I cannot figure this out: I have a square in the plane with side length $5$. $A$ and $B$ are points in the square. The coordinates of $A$ and $B$ are always integers. I want to know how many unique Euclidean distances are possible between $A$ and…
jorrebor
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Question: What theorem should I use for this geometry problem.

I have already solved this problem using trig, however I feel that their must be an easier way to solve this problem using some theorem or property of quadrilaterals that I am forgetting. Initially I tried solving this problem using strictly…
Brooks
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Unique tangent to conics

Given a conic section $C$ it is easy to prove analytically (or algebraically) that there is a unique tangent to $C$ in each point. Is there a simple synthetic proof of this fact? References are also welcome.
user23365
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Geometry problem involving orthocentre and midpoint of sides.

Let $AA_1, BB_1, CC_1$ be the altitudes of $\Delta ABC$ and let $AB \neq BC$. If $M$ is midpoint of $BC$, $H$ the orthocentre of $\Delta ABC$ and $D$ the intersection of $BC$ and $B_1C_1$, prove that $DH \perp AM$ I was trying to come up with a…
Gerard
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Area and circles

I need help with this hard geometry problem. Given a $\triangle ABC$. M is the midpoint of AB. $k_1$ and $k_2$ are the circles with diameters AC and BC respectively. $A_1$ and $A_2$ are the midpoints of the two arcs AC of $k_1$. $B_1$ and $B_2$ are…
Adam
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Hint on solving a problem in Euclidean geometry

How do i prove the following problem: If a quadrilateral has sides of length $a$, $b$, $c$, and $d$, prove that its area $S$ satisfies the following inequality $$4S\leq (a+c)(b+d)$$ with equality holding only for rectangles. Hint: Twice the area of…
user
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What is the area of polygon DHEIFG?

In any triangle ABC, D, E and F are the midpoints of AB, BC and AC and from which perpendiculars are dropped on sides AB, BC and AC.The area of triangle ABC is S. Find the area of the polygon DHEIFG in terms of S.
pirsquare
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Parallelogram from arbitrary quadrilateral

Let $ABCD$ be an arbitrary quadrilateral, four perpendicular bisector of $AB, BC, CD, DA$ form quadrilateral $A_1B_1C_1D_1$. Let $MNPQ$ be the Varignon parallelogram of $A_1B_1C_1D_1$. Let $M', N', P', Q'$ be the reflection of $M, N, P, Q$ in $AB,…
Oai Thanh Đào
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Area and orthocenter

I need help with the following proble. Given a triangle ABC with orthocenter H and altitudes $CM= 2 \sqrt 2$ and $AN=3$. If H divides the altitude BP into segments with ratio 5:1, i.e.BH:HP=5:1, find the area of $\triangle ABC$.
Adam
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2 points in n-dimensional space

Given 2 points $p_1=(x_1^1, x_2^1, ..., x_n^1)$ and $p_2=(x_1^2, x_2^2, ..., x_n^2)$ in $n$-dimensional Euclidean space, how would you define the straight-line from $p_1$ to $p_2$ with these 2 points being the endpoints of the line. There are a few…
mojo1mojo2
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Normal to a plane

My textbook makes a quick, unproven claim that if a vector P is orthogonal to /one/ vector that lies on a plane G, then P is normal to the plane. My question is, is this a mistake? Wouldn't you need P to be orthogonal to two vectors on G not…
r34
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